where we have used (REF ) with \(A_{ij}\rightarrow 2\kappa ^2 t\delta _{ij}, {\bf b}\rightarrow ({\bf x}-{\bf y})\)  to evaluate the Gaussian
integral in the first line, and \(K_\nu (z)\)  is the so-called Macdonald function, see the formula 3.471.9 of [1]}.
In particular, for \(N=2\)  and \(m\rightarrow 0\)  we have from the expansion 3.471.9 of [1]}
\(\langle V({\bf x})\,V({\bf y})\rangle =\frac{1}{2\pi \kappa ^2}K_{0}\left(\frac{m}{\kappa }|{\bf x}-{\bf y}|\right)\approx -\frac{1}{2\pi \kappa ^2}\ln {\left[\frac{|{\bf x}-{\bf y}|}{2\kappa /m}\right]}, \quad |{\bf x}-{\bf y}|\ll \frac{\kappa }{m}.\) 
